CSSS/POLS 512

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.title[
# CSSS/POLS 512
]
.subtitle[
## Lab 3: Modeling Stationary Time Series — Estimation, Selection, and Cross-Validation
]
.author[
### Ramses Llobet
]
.date[
### Spring 2026
]

---

# Today's Plan

.pull-left[
**Part 1: The applied example + diagnostics**
- UK road deaths & the 1983 seatbelt law
- Diagnostic workflow (quick review)

**Part 2: Estimating ARMA with covariates**
- `arima()` + `xreg` refresher
- **Impact effect vs. long-run equilibrium**
- Visualizing the dynamic response
]

.pull-right[
**Part 3: Model selection**
- AIC / BIC / MSE (brief)
- The full 11-candidate grid

**Part 4: Cross-validation**
- Why standard CV fails for time series
- Expanding vs. sliding windows
- Full CV grid + best model

**Part 5: Predicted values & intervals**
]

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# Part 1: The Case Study

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# UK Road Deaths and the 1983 Seatbelt Law

**Research question:** How large was the effect of the seatbelt law on road deaths, accounting for **seasonality** and **autoregressive dynamics**?

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# Diagnostic Workflow (Review from Lab 2)

Before estimation, apply the Box-Jenkins workflow:

.pull-left[
**1. Plot** the series — visible trend, seasonality, level shifts?

**2. STL decomposition** — separate trend + seasonal + remainder.

**3. ACF/PACF** on the raw series — identify candidate AR/MA structure.

**4. Clean the series** — `lm()` on trend + month dummies + law — and re-examine the ACF/PACF of the residuals.
]

.pull-right[
**What we find for UK deaths:**

- Strong **seasonal cycle** at lag 12
- **Trend** + policy shift at 1983
- After cleaning: ACF tails off, PACF significant at lags 1 and 2
- → **AR(1) or AR(2)** on the residual dynamics
]

**Strategy:** combine the covariates (law + month dummies) with an ARMA error structure in a **single joint model** via `arima()` with `xreg`.

.footnote[*Code: `Lab3.Rmd` → Sections 1.1–1.2*]

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# Part 2: ARMA with Covariates

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# Regression with ARMA Errors

`$$y_t = \underbrace{\boldsymbol{x}_t \boldsymbol{\beta}}_{\text{covariates}} + \underbrace{\eta_t}_{\text{ARMA errors}}$$`

In R:

```r
Arima(deaths, order = c(2, 0, 0), xreg = cbind(law, month_dummies))
```

`arima()` estimates `$\boldsymbol{\beta}$` and the ARMA parameters **jointly by MLE**. OLS with a lagged DV would be biased; `arima()` handles the serial correlation correctly.

**The key interpretive challenge:** the coefficient on `law` is the *immediate* effect. But AR persistence means the effect **propagates** through time before the series settles.

.footnote[*Code: `Lab3.Rmd` → Sections 2.1–2.4*]

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# Impact Effect vs. Long-Run Equilibrium

For **AR(`$p$`)** errors, the equilibrium mean of the series is:

`$$\mathbb{E}(y_t) = \frac{\boldsymbol{x}_t \boldsymbol{\beta}}{1 - \sum_{k=1}^{p} \phi_k}
\qquad\Longrightarrow\qquad
\text{Long-run effect} = \frac{\hat{\beta}}{1 - \sum_{k=1}^{p} \hat{\phi}_k}$$`

.pull-left[
**Two quantities that matter:**

- **Impact** `$= \hat{\beta}$` (first-period change)
- **Long-run** `$= \hat{\beta} \cdot \frac{1}{1 - \sum\hat{\phi}_k}$` (final equilibrium shift)
]

.pull-right[
**For UK deaths AR(2)** (`$\hat{\phi}_1 \approx 0.47$`, `$\hat{\phi}_2 \approx 0.27$`):

- Impact: `$\approx -348$` deaths/month
- Multiplier: `$\approx 3.86$`
- Long-run: `$\approx -1340$` deaths/month
]

The impact is what happens in month 1; the long-run effect is where the series ends up once all dynamics have settled.

.footnote[*Code: `Lab3.Rmd` → Section 2.5*]

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# The Dynamic Response to the Seatbelt Law

Sharp drop at month 1 (impact) → gradual adjustment → new equilibrium.

.footnote[*Code: `Lab3.Rmd` → Section 2.5 (step-response + horizon choice)*]

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# Part 3: Model Selection

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# Goodness of Fit: AIC, BIC, MSE

| Metric | What it rewards | Penalty |
|:-------|:----------------|:--------|
| **AIC** | Likelihood fit | `$+2k$` per parameter |
| **BIC** | Likelihood fit | `$+k\ln n$` (heavier) |
| **MSE** | Raw fit | **none** — always favors bigger models |

**Key facts:**

- For models fit by **MLE** (like `arima()`), AIC/BIC are the natural criteria — same likelihood the estimator already maximized
- For normal regression: `$\text{AIC} = n\ln(\text{MSE}) + 2k + C$` (constant drops out in comparisons)
- AIC favors predictive accuracy; BIC favors parsimony
- When they disagree → prefer the simpler model (Occam's razor)

**Practical workflow:** fit a grid of candidates, compare AIC + residual diagnostics (Ljung-Box, Jarque-Bera), pick the most parsimonious model whose residuals pass.

.footnote[*Code: `Lab3.Rmd` → Sections 3.1–3.3*]

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# Part 4: Cross-Validation

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# Why Standard CV Fails for Time Series

In cross-sectional data, we randomly split into train/test sets.

But in time series, **we cannot predict the past from the future** — random splits break the temporal ordering.

### Standard hold-out CV

Random train/test split → breaks the time order.

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# `$k$`-fold CV

Shuffling observations into folds still violates temporal ordering. **What we need instead:** a scheme where training always comes **before** testing.

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# Expanding Window

Training set grows as we move forward; test set is always after the training set.

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# Sliding (Rolling) Window

Fixed-length training window; useful when the process may change over time.

.footnote[*Code: `Lab3.Rmd` → Section 4.2–4.3 (manual loop) and 4.4 (`tsCV()`)*]

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# Full CV Grid: 11 Candidate Models

.footnote[*Code: `Lab3.Rmd` → Section 4.4*]

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# Part 5: Predicted Values & Intervals

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# Forecasting with `forecast()`

Point forecast + prediction intervals grow with horizon to reflect accumulating uncertainty.

.footnote[*Code: `Lab3.Rmd` → Section 5.1–5.3 (single forecast) and 5.4 (two-scenario comparison)*]

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# Wrap-Up

**Today we covered:**
- ARMA estimation with covariates via `Arima()` + `xreg` (MLE, not OLS)
- Interpreting dynamics: **impact vs. long-run equilibrium** (AR(1) → AR(p))
- Model selection: AIC/BIC (MLE natural) + residual diagnostics
- Time series cross-validation: sliding/expanding windows, manual loop + `tsCV()`
- Forecasting with `xreg`: point forecasts + prediction intervals

**Coming up next (Lab 4):**
- Non-stationary time series and ARIMA(p, **d**, q)
- **Counterfactual forecasting** with parametric simulation
- **Interrupted time series (ITS)** for causal inference

**Self-study:**
- Box-Steffensmeier et al. (2014), Ch. 2.4–2.5
- Hyndman & Athanasopoulos, [FPP3 Ch. 5](https://otexts.com/fpp3/toolbox.html) (time series CV)

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# Questions?